1_srivastava.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 5, No. 2, 2012, 97-107
ISSN 1307-5543 – www.ejpam.com

Probabilistic Proofs of Some Relationships Between the Bernoulli

and Euler Polynomials

H. M. Srivastava1,∗, Christophe Vignat2

1 Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W
3R4, Canada
2 Information Theory Laboratory, École Polytechnique Fédérale de Lausanne,
Station 14, 1015 Lausanne, Switzerland

Abstract. The main purpose of this article is to provide probabilistic proofs of the relationships be-
tween the generalized Bernoulli (or Nörlund) polynomials B(α)n (x) and the generalized Euler polyno-
mials E(α)n (x) of (real or complex) order α and degree n in x , which were proved recently by Srivastava
and Pintér [11]. Some other approaches to these relationships and their seemingly interesting gener-
alizations are also investigated.

2010 Mathematics Subject Classifications: 11B68, 60E07, 11B83, 62E15.

Key Words and Phrases: Bernoulli polynomials; Euler polynomials; Generating functions; Probabilis-
tic proofs; Umbral calculus.

1. Introduction, Definitions and Preliminaries

Throughout this paper, we use the following standard notations:

N := {1,2,3, · · · }, N0 := {0,1,2,3, · · · }= N∪ {0}

and
Z− := {−1,−2,−3, · · · } = Z−0 \ {0}.

Also, as usual, Z denotes the set of integers, R denotes the set of real numbers and C denotes
the set of complex numbers.

The classical Bernoulli polynomials Bn (x) and the classical Euler polynomials En (x), to-
gether with their familiar generalizations B(α)n (x) and E(α)n (x) of (real or complex) order α,
are usually defined by means of the following generating functions (see, for details, [2, Vol.

∗Corresponding author.

Email addresses: harimsri�math.uvi
.
a (H. M. Srivastava), 
hristophe.vignat�epfl.
h (C. Vignat)

http://www.ejpam.com 97 c© 2012 EJPAM All rights reserved.



H. M. Srivastava, C. Vignat / Eur. J. Pure Appl. Math, 5 (2012), 97-107 98

III, p. 253 et seq.], [4, Section 2.8] and [9, p. 61 et seq.]; see also [1], [2, Vol. I, p. 35 et seq.],
[5], [10, p. 81 et seq.] and [8], and the references cited therein):

� t

et − 1

�α

· ex t =

∞
∑

n=0

B(α)n (x)
tn

n!
(|t| < 2π; 1α := 1) (1)

and
�

2

et + 1

�α

· ex t =

∞
∑

n=0

E(α)n (x)
tn

n!
(|t| < π; 1α := 1) , (2)

so that, obviously, the classical Bernoulli polynomials Bn(x) and the classical Euler polynomi-
als En(x) are given, respectively, by

Bn (x) := B(1)n (x) and En (x) := E(1)n (x)
�

n ∈ N0
�

. (3)

For the classical Bernoulli numbers Bn and the classical Euler numbers En, we have

Bn := Bn (0) = B(1)n (0) and En := En (0) = E(1)n (0)
�

n ∈ N0
�

, (4)

respectively.
Recently, for the generalized Bernoulli (or Nörlund) polynomials B(α)n (x) and the general-

ized Euler polynomials of order α and degree n in x , Srivastava and Pintér [11] proved the
following two theorems.

Theorem 1 (see Srivastava and Pintér [11, p. 379, Theorem 1]). The following identity holds
true:

B(α)n

�

x + y
�

=

n
∑

k=0

�

n

k

��

B(α)k

�

y
�

+
k

2
B(α−1)

k−1

�

y
�

�

En−k (x) (5)

(α ∈ C; n ∈ N0).

Theorem 2 (see Srivastava and Pintér [11, p. 380, Theorem 2]). The following identity holds
true:

E(α)n

�

x + y
�

=

n
∑

k=0

2

k+ 1

h

E(α−1)
k+1

�

y
�

− E(α)k+1

�

y
�

i

Bn−k (x) (6)

(α ∈ C; n ∈ N0).

The main objective of this sequel to the aforementioned work by Srivastava and Pintér
[11] is to provide probabilistic proofs of the relationships between the Bernoulli and Eu-
ler polynomials, which are asserted by Theorems 1 and 2. Some other approaches to the
Srivastava-Pintér identities and their seemingly interesting generalizations are also investi-
gated.



H. M. Srivastava, C. Vignat / Eur. J. Pure Appl. Math, 5 (2012), 97-107 99

2. A Set of Useful Probabilistic Tools

In this section, we recall several probabilistic tools which will be needed in Section 3
for the probabilistic proofs of Theorems 1 and 2. First of all, Sun [12] gave the following
probabilistic representation of the Bernoulli polynomials Bn(x) and the generalized Bernoulli
(or Norlünd) polynomials B(α)n (x) of order α ∈ N0. Throughout this paper, we follow the usual
convention and tacitly assume that an empty sum and an empty product are interpreted to be
0 and 1, respectively.

Lemma 1 (see Sun [12]). Given a sequence
�

Ln
	

n∈N of independent random variables, each
with the Laplace distribution 1

2
exp (−|x |) (x ∈ R), define the random variable LB by

LB =

∞
∑

k=1

Lk

2πk
. (7)

Then each of the following probabilistic representations holds true:

Bn (x) = E

��

ıLB + x −
1

2

�n�

(n ∈ N0; x ∈ R; ı2 = −1) (8)

and

B(α)n (x) = E





 

x +
α
∑

i=1

�

ıL (i)B −
1

2

�

!n

 (n ∈ N0; α ∈ N0; x ∈ R), (9)

where the random variables
n

L (i)B

o

1≤i≤α
are independent and distributed as LB in (7).

Remark 1. The symbol EX denotes the expectation operator given by

EX
�

g (X )
�

=

∫

fX (x) g (x) d x ,

where fX is the probability density of the relevant random variable X . Moreover, in the absence
of ambiguity, we will use the simple notation E.

Remark 2. The random variable LB, defined by (7) as an infinite sum of independent random
variables, may seem to be difficult to use. We, therefore, propose the following characterization,
which can be easily proved by looking at the characteristic function of the random variable LB

as defined by (7).

Lemma 2. The random variable LB in (7) follows a logistic distribution with the density given
by

fLB
(x) =

π

2
sech2 (πx) (x ∈ R). (10)

Sun [12] also derived the following formulas for the Euler polynomials En(x) and the
generalized Euler polynomials E(α)n (x) of order α ∈ N0.



H. M. Srivastava, C. Vignat / Eur. J. Pure Appl. Math, 5 (2012), 97-107 100

Lemma 3 (see Sun [12]). If the random variable LE is defined by

LE =

∞
∑

k=1

Lk

(2k− 1)π
(11)

where
�

Lk
	

k∈N are independent Laplace random variables, then each of the following probabilistic
representations holds true:

En (x) = E

��

ıLE + x −
1

2

�n�

(n ∈ N0; x ∈ R). (12)

More generally, for α ∈ C,

E(α)n (x) = E





 

x +
α
∑

i=1

�

ıL (i)E −
1

2

�

!n

 (n ∈ N0; α ∈ C; x ∈ R) (13)

where the random variables
n

L (i)E

o

1≤i≤α
are independent and distributed as LE in (11).

Remark 3. As in the case of the Bernoulli polynomials, a more convenient characterization of
the random variable LE is provided by the following lemma.

Lemma 4. The random variable LE follows the hyperbolic secant distribution

fLE
(x) = sech (πx) . (14)

Lemma 5 below provides a fundamental property of each of the random variables LB and
LE.

Lemma 5. If UB is uniformly distributed over [0,1] and independent of LB, then, for any entire
function ϕ(x) and for all x ∈ C,

E

�

ϕ

�

x + UB + ıLB −
1

2

��

= ϕ (x) . (15)

Furthermore, if UE is a Bernoulli random variable:

Pr
�

UE = 0
	

= Pr
�

UE = 1
	

=
1

2

independent of LE, then

E

�

ϕ

�

x + UE + ıLE −
1

2

��

= ϕ (x) (16)

for any entire function ϕ(x) and for all x ∈ C.



H. M. Srivastava, C. Vignat / Eur. J. Pure Appl. Math, 5 (2012), 97-107 101

Proof. It suffices to check, with

L =LB or LE and U = UB or UE ,

that

E

��

x + U + ıL −
1

2

�n�

= xn (n ∈ N0; x ∈ C). (17)

The result (17) can be easily derived by using the moment generating functions of the cor-
responding random variables. For example, in the case of the Bernoulli polynomials, we find
that

E
�

exp
�

zUB
��

=
exp (z)− 1

z
and

E

�

exp

�

z

�

ıLB −
1

2

�
��

=
z

exp (z)− 1
;

hence

E

�

exp

�

z

�

UB + ıLB −
1

2

�
��

= 1,

which demonstrates the result asserted by Lemma (5).

Remark 4. Lemma (5) expresses the fact that the independent random variables

UB and ıLB −
1

2
or

UE and ıLE −
1

2
cancel each other in the sense that any non-zero moment of their sum equals 0. We will also need
a corollary of Lemma (5) in the following form.

Lemma 6. If, for all x ∈ C,

E
�

ϕ (x + Z)
�

= E
�

ψ (x + Z)
�

with

Z = UB, UE , ıLB −
1

2
or ıLE −

1

2
,

then
ϕ (x) =ψ (x) (x ∈ C).

Proof. If, for example,

E
�

ϕ
�

x + UB
��

= E
�

ψ
�

x + UB
��

(x ∈ C),

then the result asserted by Lemma 6 follows upon setting

x 7→ x + ıLB −
1

2
(x ∈ C).

Our demonstration of Lemma 6 is thus completed.



H. M. Srivastava, C. Vignat / Eur. J. Pure Appl. Math, 5 (2012), 97-107 102

3. Probabilistic Proofs of Theorems 1 and 2

We now use the tools presented in the preceding section in order to prove the Srivastava-
Pintér identities (5) and (6).

3.1. Proof of Theorem 1.

Assuming first that α ∈ N, let us replace the variables x and y in (5) by

x + UE and y +
α
∑

i=1

U (i)B ,

respectively. The left-hand side of the Srivastava-Pintér identity (5) reads as follows:

E



B(α)n

 

x + UE + y +
α
∑

i=1

U (i)B

!

 = E
��

x + y + UE
�n �

=
1

2

�

x + y + 1
�n
+

1

2

�

x + y
�n . (18)

The same operation in the right-hand side of the Srivastava-Pintér identity (5) yields

E





n
∑

k=0

�

n

k

�

B(α)k

 

y +
α
∑

i=1

U (i)B

!

En−k
�

x + UE
�





= E







n
∑

k=0

�

n

k

�

 

y +
α
∑

i=1

�

ıL (i)B −
1

2

�

+

α
∑

i=1

U (i)B

!k

xn−k







=

n
∑

k=0

�

n

k

�

yk xn−k

=
�

x + y
�n (19)

for the first term, and

1

2

d

d y

n
∑

k=0

�

n

k

�

E







 

y +
α
∑

i=0

U (i)B +

α−1
∑

i=1

�

ıL (i)B −
1

2

�

!k

xn−k







=
1

2

d

d y

§

E

�
�

x + y + U (α)B

�n�ª

=
1

2

�
�

x + y + 1
�n−

�

x + y
�n� (20)

for the second term. By applying the assertion of Lemma 6, the observations (19) and (20),
together, conclude the proof of the Srivastava-Pintér identity (5).



H. M. Srivastava, C. Vignat / Eur. J. Pure Appl. Math, 5 (2012), 97-107 103

3.2. Proof of Theorem 2.

In (6) we replace the variables x and y by

x + UB and y +
α
∑

i=1

U (i)E ,

respectively. We thus obtain

E



E(α)n

 

x + UB + y +
α
∑

i=1

U (i)E

!



= E
��

x + y + UB
�n �

=
1

n+ 1

�
�

x + y + 1
�n+1 −

�

x + y
�n+1� (21)

for the left-hand side. For the right-hand side, we similarly obtain

E





 

E(α−1)
k+1

 

y +
α
∑

i=1

U (i)E

!

− E(α)k+1

 

y +
α
∑

i=1

U (i)E

!!

Bn−k
�

x + UB
�





=

��

1

2

�

y + 1
�k+1

+
1

2
yk+1

�

− yk+1
�

xn−k

=

�

1

2

�

y + 1
�k+1−

1

2
yk+1

�

xn−k. (22)

The derivative of the right-hand side sum in (22) is given by

n
∑

k=0

�

n

k

�

�

�

y + 1
�k − yk

�

xn−k =
�

x + y + 1
�n−

�

x + y
�n ,

which obviously coincides with the derivative of the left-hand side sum in (21). Applying the
assertion of Lemma 6 once again, we are led to the Srivastava-Pintér identity (6).

4. Further Remarks and Observations

In this concluding section, we begin by presenting several further remarks and observa-
tions concerning (for example) the scope and prospects of our probabilistic and other ap-
proaches to the Srivastava-Pintér identities (5) and (6) asserted by Theorems 1 and 2, respec-
tively.

Remark 5. Although the probabilistic proofs of Theorems 1 and 2 were given in the preceding
section only in the case when α ∈ N0, yet they can be extended appropriately to any complex-
valued parameter α, since the function α 7→ B(α)n (x) is a polynomial of degree n in α. For
example, we have

B(α)0 (x) = 1, B(α)1 (x) = x −
α

2
,



H. M. Srivastava, C. Vignat / Eur. J. Pure Appl. Math, 5 (2012), 97-107 104

B(α)2 (x) = x2− xα+
α

6
+
α (α− 1)

4
,

and so on. Hence any identity that holds true for all α ∈ N0 extends also to the whole complex
α-plane. Furthermore, the Bernoulli (or Nörlund) polynomials B(−α)n (x) (α ∈ N0) generated by

∞
∑

n=0

B(−α)n (x)
tn

n!
=

�

et − 1

t

�α

· ex t (|t| < 2π; α ∈ N0) (23)

can be expressed as follows as moments:

B(−α)n (x) = E





 

x +
α
∑

i=1

U (i)B

!n

 (α ∈ N0). (24)

Similarly, for the Euler polynomials E(−α)n (x) (α ∈ N0) generated by

∞
∑

n=0

E(−α)n (x)
tn

n!
=

�

et + 1

2

�α

· ex t (|t| < π; α ∈ N0), (25)

we have

E(−α)n (x) = E





 

x +
α
∑

i=1

U (i)E

!n

 (α ∈ N0). (26)

Remark 6. Another approach to the identities (5) and (6) for α ∈ N0 consists in proving first
their cases when α= 0, namely

B(0)n

�

x + y
�

=

n
∑

k=0

�

n

k

��

B(0)k

�

y
�

+
k

2
B(−1)

k−1

�

y
�

�

En−k (x) (27)

for the identity (5).
The special identity (27) can be checked easily, since

B(0)n (x) = xn and B(−1)
k−1

�

y
�

= E

�

�

y + UB
�k−1

�

so that the left-hand side of (27) reads

�

x + y
�n ,

while the right-hand side of (27) is given by

E

 

n
∑

k=0

�

n

k

��

yk +
k

2

�

y + UB
�k−1

�

En−k (x)

!

= En
�

x + y
�

+
1

2

�

En
�

x + y + 1
�

− En
�

x + y
��



H. M. Srivastava, C. Vignat / Eur. J. Pure Appl. Math, 5 (2012), 97-107 105

= E

�

En
�

x + y + UE
�

�

=
�

x + y
�n .

Thus, upon replacing y by

y +
α−1
∑

i=1

�

ıL (i)B −
1

2

�

,

we are led to the identity (5).

Remark 7. The approach indicated in Remark 6 suggests a generalization of the identity (5) to
the Bernoulli polynomials B(α)n (x |a) of order α ∈ N, degree n and parameter a ∈ Rα defined by
the following generating function (see [2]):

∞
∑

n=0

B(α)n (x |a)
tn

n!
= exp (x t)

α
∏

k=1

�

ak t

exp
�

ak t
�

− 1

�

. (28)

It can be easily verified that

B(α)n (x |a) = E





 

x +
α
∑

i=1

ai

�

ıL (i)B −
1

2

�

!n

 (29)

and that the case a = (1, · · · , 1) corresponds to the Bernoulli (or Nörlund) polynomials B(α)n (x)
(α ∈ N0). The Euler case reads analogously as follows:

∞
∑

n=0

E(α)n (x |a)
tn

n!
= exp (x t)

α
∏

k=1

�

2ak

exp
�

ak t
�

+ 1

�

(30)

and

E(α)n (x |a) = E





 

x +
α
∑

i=1

ai

�

ıL (i)E −
1

2

�

!n

 . (31)

Finally, we state and prove the following result.

Theorem 3. For n ∈ N0, α ∈ C, a ∈ Cα and any j (1≦ j ≦ α) such that a j 6= 0,

B(α)n

�

x + y|a
�

=

n
∑

k=0

�

n

k

��

B(α)k

�

y|a
�

+ a j
k

2
B(α−1)

k−1

�

y|a \ a j

�

�

· En−k

�

x |a j

�

(32)

and

E(α)n

�

x + y|a
�

=

n
∑

k=0

2

k+ 1

�

1

a j
E(α−1)

k+1

�

y|a \ a j

�

−
1

a j
E(α)k+1

�

y|a
�

�

· B(1)n−k

�

x |a j

�

, (33)

where
a \ a j :=

�

a1, · · · , a j−1, a j+1, · · · , aα
�

.



REFERENCES 106

Proof. Starting from the identity (5) with α = 1, if we replace x and y by

x

a j
and

y

a j
,

respectively, we obtain

E

��

x

a j
+

y

a j
+

�

ıL (
j)

B −
1

2

�
�n�

= E

� n
∑

k=0

�

n

k

�

·

��

y

a j
+

�

ıL (
j)

B −
1

2

�
�k

+
k

2

�

y

a j

�k−1��
x

a j
+ ıLE −

1

2

�n−k�

,

which, when multiplied by an
j on both sides, yields

E

��

x + y + a j

�

ıL (
j)

B −
1

2

�
�n�

= E

� n
∑

k=0

�

n

k

�

·

��

y + a j

�

ıL (
j)

B −
1

2

�
�k

+
k

2
a j yk−1

��

x + a j

�

ıLE −
1

2

�
�n−k�

. (34)

Upon replacing y by

y +
α
∑

i=1 (i 6= j)

ai

�

ıL (i)B −
1

2

�

in (34), if we evaluate the resulting expectations, we get the first assertion (32) of Theorem 3.
The second assertion (33) of Theorem 3 can indeed be proven similarly.

Remark 8. The underlying principle of the approach involved in Remarks 6 and 7 (and leading
to Theorem 3 above) is that any Bernoulli or Euler polynomial can be represented as a moment
of a shifted monomial as (for example) in (8) and (12). This can be related to the notion of
polynomials of the binomial type which appears in the theory of operator calculus (see [6]).

Remark 9. Such other approaches as the umbral-calculus approach would allow an equally
simple path to these proofs. In this connection, we refer the reader to the seminal paper by
Rota and Taylor [7], where the notion of the cancellation properties exhibited by (15) and (16)
corresponds to the notion of the inverse umbras. The paper by Gessel [3], too, provides simple
derivations of several identities for the Bernoulli polynomials by using umbral calculus.

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